if the ratio of the ages (in years) of A and B, 5 years ago was 5:7, then which of the following can be the sum of their ages 5 years from now?

Options

A. 90
B. 98
C. 92
D. 87

Answer

To solve the problem, we need to determine the present ages of A and B based on the given ratio of their ages 5 years ago, and then calculate their ages 5 years from now.

### Step-by-Step Solution:

1. **Understanding the Given Ratio:**
- The ratio of the ages of A and B, 5 years ago, was given as 5:7.
- This means that if we let the ages of A and B be represented as \(5x\) and \(7x\) respectively (where \(x\) is a common multiplier).

2. **Calculating Present Ages:**
- Since the ages were \(5x\) and \(7x\) five years ago, we need to find their present ages by adding 5 years to each of their ages.
- Present age of A = \(5x + 5\)
- Present age of B = \(7x + 5\)

3. **Calculating Ages 5 Years from Now:**
- To find their ages 5 years from now, we will add another 5 years to their present ages.
- Age of A after 5 years = \( (5x + 5) + 5 = 5x + 10 \)
- Age of B after 5 years = \( (7x + 5) + 5 = 7x + 10 \)

4. **Finding the Sum of Their Ages 5 Years from Now:**
- Now, we need to find the sum of their ages 5 years from now.
- Sum of ages = \( (5x + 10) + (7x + 10) = 12x + 20 \)

5. **Identifying Possible Values for the Sum:**
- The sum of their ages 5 years from now is \(12x + 20\).
- Since \(x\) is a positive integer, we can substitute different values for \(x\) to find possible sums.
- For example:
- If \(x = 1\), then sum = \(12(1) + 20 = 32\)
- If \(x = 2\), then sum = \(12(2) + 20 = 44\)
- If \(x = 3\), then sum = \(12(3) + 20 = 56\)
- And so on...

6. **Conclusion:**
- The possible sums of their ages 5 years from now can be expressed as \(12x + 20\), where \(x\) is a positive integer. Thus, the sums will be in the form of \(32, 44, 56, ...\) and so on.